It is not inappropriate to call it a cooperative business in the case of Rainbow or Other Avenues (or any other worker-owned business); they are worker co-ops, as opposed to consumer co-ops - though it is true that when people talk about food co-ops, they usually mean consumer co-ops, and this is probably true of the OP.

To me, a co-op grocery is a nonprofit community project, owned by the customers. There were a dozen or more of those in SF in the 70s and early 80s, e.g. the Noe Valley Community Store (which also turned into a collective before closing).